Time allowed: 20 minutes.
1. (3 marks)
It is known that the vector
⎛
⎜
⎜
⎝
2
0
1
k
⎞
⎟
⎟
⎠
is in span
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎛
⎜
⎜
⎝
1
2
3
4
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
3
6
9
12
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
3
3
3
0
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
1
−
1
−
4
−
8
⎞
⎟
⎟
⎠
⎫
⎪
⎪
⎬
⎪
⎪
⎭
.
(a) Determine the value of
k
.
(b) Hence or otherwise, for the value of
k
determined in part (a) above, ﬁnd values of
λ
1
,λ
2
3
4
such that:
⎛
⎜
⎜
⎝
6
0
3
3
k
⎞
⎟
⎟
⎠
=
λ
1
⎛
⎜
⎜
⎝
1
2
3
4
⎞
⎟
⎟
⎠
+
λ
2
⎛
⎜
⎜
⎝
3
6
9
12
⎞
⎟
⎟
⎠
+
λ
3
⎛
⎜
⎜
⎝
3
3
3
0
⎞
⎟
⎟
⎠
+
λ
4
⎛
⎜
⎜
⎝
1
−
1
−
4
−
8
⎞
⎟
⎟
⎠
.
2. (3 marks)
We would like to ﬁt the quadratic curve
y
=
β
1
+
β
2
x
+
β
3
x
2
to a set of points (
x
1
,y
1
)
,
(
x
2
2
)
, ... ,
(
x
n
n
)
by the method of least squares.
(a) Write down the normal equations in matrix form.
(b) For this dataset,
n
=12
,
∑
x
i
=22
,
∑
x
2
i
=11
,
∑
x
3
i
=6
,
∑
x
4
i
=1
,
∑
x
5
i
=0
.
5
,
∑
y
i
=45
,
∑
x
i
y
i
=38
,
∑
x
2
i
y
i
,
∑
x
i
y
2
i
=35
,
∑
x
2
i
y
2
i
= 58.
Show that
β
1
,β
2
= 2 and
β
3
=
−
1.
3. (4 marks)
Suppose that
˜
v
1
=
⎛
⎜
⎜
⎝
1
0
0
1
⎞
⎟
⎟
⎠
,
˜
v
2
=
⎛
⎜
⎜
⎝
1
0
1
0
⎞
⎟
⎟
⎠
,
˜
v
3
=
⎛
⎜
⎜
⎝
1
1
1
1
⎞
⎟
⎟
⎠
and
π
= span(
˜
v
1
,
˜
v
2
).
(a) Find an orthogonal basis for
π
.
(b) Hence, or otherwise, ﬁnd proj
π
˜
v
3
.